Quasi-decomposition of abelian groups and Baer's lemma
Let \(A\) be a finite rank torsion-free abelian group. An abelian group \(G\) is called quasi-\(A\)-projective if \(G\) is quasi-isomorphic to a direct summand of a finite direct sum of copies of \(A\). For any group \(C\), \(S_ A(C)\) is the subgroup of \(C\) generated by \(\{\varphi(A):\varphi\in \text{Hom}(A,C)\}\). The abelian group \(A\) is said to have the quasi-Baer- splitting property if every exact sequence \(0 \to B @>\alpha>> C @>\beta>> G \to 0\) where \(G\) is quasi-\(A\)-projective and \(C\) is quasi equal to \(\alpha(B) + S_ A(C)\) is quasi-splitting. The author gives necessary and sufficient conditions for \(A\) to have the quasi-Baer- splitting property in terms of the strongly indecomposable summands \(A_ 1,\dots,A_ n\) that show up in a fixed quasi-decomposition of \(A\): These groups \(A_ i\) must have incomparable socles, which properly is defined in terms of the socles \(S_{A_ j}(X)\) of subgroups \(X\) of \(A_ i\) for various \(i\) and \(j\). In particular, an almost completely decomposable group \(A\) will have the quasi-Baer-splitting property exactly when all its critical types are pair-wise incomparable.
scientific article; zbMATH DE number 219330
scientific article